Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 78

Chapter 2, Problem 78

Solve each rational inequality. Give the solution set in interval notation. 4/(x+1)<2/(x+3)

Verified step by step guidance

Start by writing the inequality clearly: $\frac{4}{x+1} < \frac{2}{x+3}$.

Bring all terms to one side to have zero on the other side: $\frac{4}{x+1} - \frac{2}{x+3} < 0$.

Find a common denominator, which is $(x+1)(x+3)$, and combine the fractions: $\frac{4(x+3) - 2(x+1)}{(x+1)(x+3)} < 0$.

Simplify the numerator: $4(x+3) - 2(x+1) = 4x + 12 - 2x - 2 = 2x + 10$, so the inequality becomes $\frac{2x + 10}{(x+1)(x+3)} < 0$.

Determine the critical points by setting numerator and denominator equal to zero: numerator $2x + 10 = 0$ gives $x = -5$, denominator factors $x+1=0$ and $x+3=0$ give $x = -1$ and $x = -3$. Use these points to test intervals on the number line to find where the expression is less than zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while considering restrictions where the denominator is zero to avoid undefined expressions.

Finding a Common Denominator and Combining Terms

To solve rational inequalities, it is often necessary to rewrite both sides with a common denominator. This allows combining the inequality into a single rational expression, making it easier to analyze the sign of the expression.

Sign Analysis and Interval Testing

After combining terms, determine where the numerator and denominator are zero to find critical points. Use these points to divide the number line into intervals and test each interval to see where the inequality holds, considering domain restrictions.

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Solve each rational inequality. Give the solution set in interval notation. (x+3)/(2x-5)≤1

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